Asymptotic behavior of an affine random recursion in Zpk defined by a matrix with an eigenvalue of size 1

In this paper we study the rate of convergence of the Markov chain Xn+1=AXn+Bn(modp), where AA is an integer invertible matrix, and {Bn}n is a sequence of independent and identically distributed integer vectors. If AA has an eigenvalue of size 1, then n=O(p2)n=O(p2) steps are necessary and sufficient to have Xn sampling from a nearly uniform distribution.